Abstract
This paper focuses on signal processing tasks in which the signal is transformed from the signal space to a higher dimensional coefficient space (also called phase space) using a continuous frame, processed in the coefficient space, and synthesized to an output signal. We show how to approximate such methods, termed phase space signal processing methods, using a Monte Carlo method. As opposed to standard discretizations of continuous frames, based on sampling discrete frames from the continuous system, the proposed Monte Carlo method is directly a quadrature approximation of the continuous frame. We show that the Monte Carlo method allows working with highly redundant continuous frames, since the number of samples required for a certain accuracy is proportional to the dimension of the signal space, and not to the dimension of the phase space. Moreover, even though the continuous frame is highly redundant, the Monte Carlo samples are spread uniformly, and hence represent the coefficient space more faithfully than standard frame discretizations.
Highlights
We consider signal processing tasks based on continuous frames [1,46]
Mote Carlo signal processing in phase space We study a Monte Carlo approximation of signal processing in phase space based on the pipelines (5) and (6)
When considering discrete signals of resolution/dimension M, embedded√in the continuous signal space, the error in the stochastic method is of order O( M/K ), where K is the number of Monte Carlo samples
Summary
We consider signal processing tasks based on continuous frames [1,46]. The linear operator T models a global change of the signal in the feature space, while the nonlinearity r allows modifying the feature coefficients term-by-term with respect to their values. E.g., wavelet shrinkage denoising [14,15], and Shearlet denoising [29], the linear operator is trivial T = I and r is a nonlinearity that attenuates low values. An example application is shearlet or curevelet based Radon transform inversion [9, 11,20], where T is a multiplicative operator, r is thresholding, analysis is done by the curvelet/shearlet frame, and synthesis is done by some modified curvelet/shearlet frame. In this paper we study quadrature discretizations of (1) based on random samples
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